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Financial Integration, Savings Gluts, and Asset Price Booms

Felix Zhiyu Feng, Will Jianyu Lu and Caroline H. Zhu

Abstract

Capital outflows after financial integration can lead to simultaneous increases in the national savings rate and asset prices of an economy with substantial financing costs. Under autarky, firms invest in risky capital while facing a borrowing constraint that creates a need for precautionary savings. Financial integration provides firms with access to foreign risk-free assets and results in two effects: a substitution effect, whereby firms divert some investments to foreign assets and cause capital outflows; and a wealth effect, whereby they grow richer in equilibrium and thus demand more domestic capital. Savings gluts and asset price booms occur when the wealth effect dominates.

JEL Classification: G11; G15; F36; E22

Corresponding author: Felix Zhiyu Feng, Michael G. Foster School of Business, University of Washington, Seattle, WA, USA, E-mail:

Article note: We thank Burkhard Schipper (the editor), Attila Ambrus, Barney Hartman-Glaser, Joseph Kaboski, Vincenzo Quadrini, Adriano Rampini, Curtis Taylor, S. Viswanathan, Daniel Yi Xu, seminar participants at Duke University and University of Notre Dame for helpful comments. This paper was previously titled “Savings Gluts and Asset Price Booms in Emerging Economies”. All errors are our own.


Appendix

A. Proofs

In this Appendix we first detail the system of equations that characterize land price P and firm's portfolio choice α.

Financial Integration: in this equilibrium, firms choose optimal portfolio according to

(14)α=μPrP2βσ2.

Firms' net worth follows a geometric Brownian motion:

dWtWt=[(μrP)2βσ2+r]dt+μPrβ(σP)dZt.

Define λ=μrPσ as the Sharp ratio from investing in land. λ must be larger than 0 in equilibrium, otherwise firms will hold only risk-free bonds. Substituting λ into the dynamics of firm net worth yields

(15)dWtWt=[λ2β+r]dt+λβdZt.

Combining (14) and (15) into firm's value function yields

ρV=(λ22β+r)WtV(Wt).

The solution is V(Wt)=C1Wγ, where γ=ρ(λ22β+r)1=2ρβλ2+2rβ. The coefficient C1 and payout boundary can be pinned down by matching the refinancing boundary condition V(W¯)=γC1W¯γ1=1+ζ and the payout boundary condition V(W¯)=γC1W¯γ1=1.

C1=1+ζγW¯1γ
W¯=(1+ζ)11γW¯

For the refinancing boundary W, substituting V(W)=C1Wγ into boundary condition (7) yields

C1γW¯=ϕ+(1+ζ)W¯

and

γC1γ1W¯=1+ζ.

Together, one can solve for W¯

W¯=γϕ(1γ)(1+ζ).

Notice that βWV(Wt)V(Wt)=1γ, which implies the following relationship:

(16)2rβ2+[2(ρr)+λ2]βλ2=0

Define Φ(β,λ2) as the left hand side (LHS) quadratic function of this equation. Φ=0 therefore determines β given λ, or equivalently, given P. The power of the value function γ is given by γ=1β.

The geometric Brownian motion dWtWt=μWdt+σWdZt with two reflecting barriers W¯ and W¯ has a stationary distribution. Define η=2μWσW2. The stationary distribution is characterized by the density function

(17)f(W)=η1W¯η1W¯η1Wη2.

Finally, PI is the solution to the market clearing condition

(18)αE(W)=P,

where

E(W)=(η1)η(W¯η1W¯η1)(W¯ηW¯η).

Combining Equations (14 –16) and (18) solves for PI.

Autarky: in this equilibrium, firms invest all their net worth in land. Therefore, dWt=1PtWtdAt when Wt is between the refinancing and payout boundaries. The value function satisfies

ρV=(μPβσ22P2)WtV(Wt).

Therefore, V(Wt)=C1Wγ, where γ=2ρP22μPσ2β(0,1) and γ=1β implies

(19)σ2β2[2μP+σ2]β+2P(μρP)=0.

The dynamics of firm net worth follow another geometric Brownian motion:

(20)dWtWt=μPdt+σPdZt.

There is a stationary distribution for Wt, and its density function is given by (17). The density function implies the aggregate net worth E(W), which pins down PA from the solution to the market clearing condition:

(21)E(W)=P.

Combining Equations (19–21) solves for PA.

Proof of Lemma 1. Take (16) as a function of β. The function has one negative root and one positive root. The left hand side is smaller than 0 when β=0 and larger than 0 when β=1, hence there must be a unique solution β such that 0<β<1, which implies 0<γ<1.

For the autarky equilibrium, notice that μρPA>0, since μ/ρ is the price of capital when σ=0, i.e. when investment in capital is risk-free. The left hand side of (19) is smaller than 0 when β=1, and therefore (19) has one root larger than 1 and one smaller than 1, that is, there is a unique β such that 0<β<1. □

Proof of Proposition 1. To show the relationship between PI and PA, we start by showing that PA coincides with the solution of PI when α=1. Next, we show that PI is an increasing function of α. By limited enforcement, α<1 in equilibrium under financial integration. Therefore it must be that PI>PA.

Equation (15) implies the geometric Brownian motion of firm net worth Wt satisfies μW=λ2β+r and σW=λβ plus the lower boundary W¯ and endogenous upper boundary W¯ pinned down by matching the boundary conditions. Notice that W¯ is proportional to W¯, which is not surprising given the property of power functions. This reduces the ratio of W¯ηW¯η and W¯η1W¯η1 into a linear function of W¯. Using the relationship between β and λ derived in (16), standard algebra yields that the market clearing condition can be written as

λ2+(rρ)2(λ2+rβ)α=((1+ζ)1+1β1(1+ζ)1β1)W¯.

Notice that from (16), λ2=2rβ2+2(ρr)β1β. Substituting this into the simplified market clearing condition yields

2(rρ)2rβ2+(1ρ)(β1)4(rρ)2rβ22rβα((1+ζ)1+1β1(1+ζ)1β1)W¯=0.

Define the left hand side of this equation as Ψ(β,α), which is now only a function of two variables: firms' effective risk aversion β and their portfolio choice α. The following lemma establishes the link between the autarky equilibrium and the financial integration equilibrium through Ψ(β,α):

Lemma A1. PA is the solution to Ψ(β,1)=0.

Proof: Let α=1, then 1=μPrP2βσ2 implies r=μPβσ2P2. Plugging this back into (16) implies 2ρP22μ(1β)P+σ2β(1β)=0, which, after rearranging terms, is the same as Equation (19). Substituting (19) into (15) yields μW=μ/P and σW=σ/P for the geometric Brownian motion of the dynamics of Wt, which is identical to dWt under autarky. Notice that E(W) is defined through η as a function of μW and σW only. This implies that the market clearing condition (18) when α=1 is identical to the market clearing condition for the autarky equilibrium. Therefore, PA must equal to PI when α=1.

Now comparing PI and PA is equal to comparing the different solutions of Ψ(β,α)=0 in terms of β and α. By the chain rule, Pα=Pβλα. The signs of the two terms can be determined separately in the following two lemmas:

Lemma A2. Pβ<0.

Proof: By the chain rule, Pβ=Pλλβ. The definition of λ=μrPσ implies Pλ<0. Rearranging the terms of Equation (16) yields λ2=2(rρ)2rβ2β1 and λβ=12λ[2(ρrβ)(β1)2]12>0, sinceρ>rand β<1. Therefore Pβ<0.

Lemma A3. βα>0 under certain parameter conditions.

Proof: from Ψ(β,α)=0, one can solve for α as a function of β:

α=2((1+ζ)1+1β1(1+ζ)1β1)4(rρ)2rβ22rβ2(rρ)2rβ2+(1ρ)(β1)W¯.

Taking logarithms on both sides of the equation gives

lnα=ln2W¯+ln((1+ζ)1+1β1)ln((1+ζ)1β1)+ln(λ2+rβ)ln(λ2+(rρ)).

Differentiating this equation with respect to β yields

αβ=A+B+C,

where

A=(1+1/β)ϕ(1+ζ)1/β(1+ζ)1+1β1+(1/β)(1+ζ)1/β(1+ζ)1β1,
B=L+rλ2+rβ,
C=Lλ2+(rρ),

and Lβ(λ2).

There are three parts to characterizing the derivative of α on β: Part A measures the change of payment boundary W¯, a positive term because lower β implies weaker risk aversion, which lowers the payment boundary; Part B comes from the change of η due to the drift term μW, which is smaller when α is lower; and Part C is the wealth effect reflected by the change of η due to the diffusion term σW, which is also smaller when α is lower.

Notice that part A is bounded by

2ϕ(1+ζ)(1+ζ)211(1+ζ)2(1+ζ)=2ϕ(1+ζ)21(ζ+2)(ζ+1).

Part A reaches this upper bound, defined as A¯, when β=1. A sufficient condition for βα>0 is therefore

(22)L(rβ+ρr)r(λ2+rβ)(λ2+(rρ))A¯.

First, A¯(λ2+rβ)(λ2+(rρ))A¯λ2(λ2+rβ) and L=β(λ2)=2(ρrβ)(β1)2. Therefore, the inequality of (22) is true as long as

2(ρrβ)(β1)2(rβ+ρr)A¯λ2(λ2+rβ)+r;
2(ρrβ)(rβ+ρr)A¯λ2(λ2+rβ)(β1)2+r;
2ρ((rβ+ρr)A¯[2(rρ)4rβ2+rβ].

In every step above, the left hand side is made larger and right hand side made smaller and rβ0. Finally, since β<1, the last inequality above implies the following sufficient condition under which βα>0 is satisfied:

3r+(2r1)ρρrA¯.

Combining Lemmas A2 and A3 implies Pα=Pββα<0. By Lemma A1, PAcoincides with the solution where α=1. α<1 in the financial integration equilibrium, and therefore it must be that PA<PI. □

Proposition 1 implies the following corollary.

Corollary 1: βIn<βAu.

Proof: From (16), β is an increasing function in λ. λ is a decreasing function in P. Therefore given PA<PI, this implies βIn<βAu. This result is intuitive as precautionary savings with the risk-free asset implies that firms are less risk averse. This Corollary basically states that the wealth effect comes from the reduction of effective risk aversion.

Proof of Proposition 2:

Proof: by the definition of savings rate,

SIEI(W)μ+(1+r(1α))EI(W)

and

SAEA(W)μ+EA(W).

Suppose SI<SA. Because EI(W)>EA(W), it must be that

EI(W)μ+(1+r(1α))EI(W)<EA(W)μ+EA(W);
μEI(W)+EA(W)EI(W)<μEA(W)+(1+r(1α))EI(W)EA(W).

That is,

μ(EI(W)EA(W))<r(1α)EI(W)EA(W).

Dividing both sides by EA(W) yields

μ(EI(W)EA(W)1)<r(1α)EI(W).

First, EI(W)EA(W)=PIαPA, and therefore μPIαPA<r(1α)EI(W). That is,

rα(1α)EI(W)>μ,

or EI(W)>4μr. Recall that

E(W)=(η1)η(W¯η1W̲η1)(W¯ηW¯η)<(1+ζ)1+1β1(1+ζ)1β1.

Therefore,

(1+ζ)1+1β1(1+ζ)1β1>4μr.

The left hand side has a minimum 1+ζ when β0. Therefore, for SI<SA, a necessary condition is

(23)(1+ζ)r>4μ.

When condition (23) is violated, SI<SA cannot be true. i.e., when Condition 2, (1+ζ)r<4μ, is satisfied, it must be that SI>SA. □

Proof of Proposition 3. The firm's value function solves

(24)ρV(Wt)=maxαt1[(μPr)αtWt+rWt]V(Wt)+12αt2P2σ2Wt2V(Wt),

subject to the market clearing condition

αtWtdF(W)=P.

The market clearing condition implies P=P(α) as an explicit function of α. The first order condition of HJB Equation (24) with respect to α is

(μPr)WVσ2P2(α)αW2VμαP2P(α)WV+α2P3σ2P(α)=0.

The first two terms are the first order derivative of V with respect to α taking P as a constant. Setting them equal to zero gives αI, the solution of α to the decentralized economy where firms take the equilibrium price P as given. The third and fourth terms come from the additional market clearing constraint for solving α*, that firms take into account the effect of choosing α on the equilibrium price P. Using the same procedure of refinement as in Section III, we look for the solution where V=βV/W for some β. Substituting out V and simplifying the first order condition implies

(25)(μPr)βσ2P2αμαP2P+βα2P3σ2P=0.

Let A(α)(μPr)βσ2P2α. Equation (25) defines P as a function of α:

P(α)|α=α*=A(α*)P3α*(β*α*σ2μ).

Notice that, A(αI)=0 since αI is the solution of setting A(α)=0. Suppose that α*>αI. The proof of Proposition 1 shows that P<0 under Condition 1, implying that β*α*σ2μ<0. However, from Equation (19), algebra shows that βIαI>μσ2. According to Lemma A3, βα is increasing in α, and therefore β*α*>βIαI>μσ2, a contradiction. Then, it must be that α*<αI. □

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Received: 2018-03-29
Accepted: 2020-04-04
Published Online: 2020-08-07

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